Mathematical model for evaluation of velocity profile effects on cross-sectional concentration with application to X-ray imaging

Abstract

The goal of this dissertation is to develop mathematical models and numerical tools to analyze and quantify mean transit times of flow within a cylindrical tube from data that is acquired using cross-sectional sampling methods. Typically, indicator concentration curves in physiologic systems have been measured using flow-proportional sampling techniques. However, with the advent of improved temporal and spatial resolution imaging systems, concentration curves from vessels within living organs can be obtained. However, these curves are generally cross-sectionally sampled and thus do not fulfill the requisite assumptions of conventional indicator dilution theory. The most significant practical ramification is that generally, the mean transit time of a cross-sectional concentration curve does not equal that of a flow-sampled concentration curve or may fail to exist at all. Thus, here we have extended the application of indicator dilution theory by examining a class of model flow systems for which it is possible to obtain estimates of the flow parameters (including mean transit time) within the system using cross-sectional concentration curves. The model flow system consisted of a straight cylindrical tube perfused with steady flow having a general power-law (α) velocity profile. Measurement of upstream and downstream cross-sectional concentration curves was considered. Several new results were obtained. First, the existence and general form of a unique cross-sectional transport function between these two sites was established for 0 ≤ α < 1. This transport function has an analytic form consisting of the sum of a leading sharp delta function followed by a decreasing, continuous function that is shown to be the unique solution of a Volterra integral equation involving the power-law velocity profile. Two methods for obtaining the cross-sectional transport function from upstream and downstream cross-sectional concentration curves were developed. The damped least squares deconvolution method was investigated as well as numerical implementation of the solution of the Volterra integral equation. Finally, the relationship between the mean transit time of the cross-sectional transport function and the flow-sampled mean transit time (given by volume/flow) was obtained. The utility of these approaches was investigated via simulations and an experiment. The simulations revealed that even in the presence of noise, the methods produce reasonable estimates of the underlying parameters of the model system. The experiment demonstrated the practical utility of the numerical methods. In both cases, the underlying methodology was to simulate or measure upstream and downstream cross-sectional concentration curves which were then used to determine the cross-sectional transport function using both the damped least squares and the analytic solution methods. The mean transit time of the estimated cross-sectional transport function was then calculated and rescaled to obtain an estimate of the flow-sampled mean transit time. Further research will be needed to determine the applicability of these methods to concentration curves obtained from flow images of vessels in living organs.